Subgroups — C₄₁×#2543 #19106323

Order 5248.

724 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#121×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#121×C₄₁	C₂	1	2	yes	maximal, complemented
3	C₂×C₄×D₄×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₁×#398	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#397	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₁×#394	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₁×#392	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₁×#388	C₂	1	2	yes	maximal, complemented
9	C₄₁×#386	C₂	1	2	yes	maximal, complemented
10	C₄₁×#386	C₂	1	2	yes	maximal, complemented
11	C₂×#95×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₄×C₄₁	C₂²	1	2	yes	—
13	C₂²×D₄×C₄₁	C₂²	1	2	yes	—
14	C₂²×D₄×C₄₁	C₂²	1	1	yes	characteristic
15	C₂²×D₄×C₄₁	C₂²	1	1	yes	characteristic
16	C₂³×C₄×C₄₁	C₂²	1	2	yes	—
17	C₂³×C₄×C₄₁	C₂²	1	1	yes	characteristic
18	C₂³×C₄×C₄₁	C₂²	1	1	yes	characteristic
19	#121×C₄₁	—	2	8	no	—
20	#121×C₄₁	—	2	4	no	—
21	C₄×D₄×C₄₁	—	2	4	no	—
22	C₂×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
24	C₂×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
26	C₂×C₂²⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
27	C₂×C₂²⋊C₄×C₄₁	C₂²	1	2	yes	—
28	C₂×C₂²⋊C₄×C₄₁	C₂²	1	2	yes	—
29	C₂×C₂²⋊C₄×C₄₁	C₂²	1	2	yes	—
30	C₂×C₂²⋊C₄×C₄₁	C₂²	1	2	yes	—
31	C₂×C₂²⋊C₄×C₄₁	C₂²	1	2	yes	—
32	C₂×C₄²×C₄₁	C₂²	1	1	yes	characteristic
33	#95×C₄₁	C₂²	1	2	yes	—
34	#95×C₄₁	C₂²	1	1	yes	characteristic
35	#95×C₄₁	C₂²	1	1	yes	characteristic
36	#95×C₄₁	C₂²	1	2	yes	—
37	#95×C₄₁	C₂²	1	2	yes	—
38	C₂⁴×C₄₁	C₂³	1	2	yes	—
39	C₂⁴×C₄₁	C₂³	1	1	yes	characteristic
40	C₂⁴×C₄₁	C₂³	1	1	yes	characteristic
41	C₂×D₄×C₄₁	—	2	2	no	—
42	C₂×D₄×C₄₁	—	2	2	no	—
43	C₂×D₄×C₄₁	—	2	4	no	—
44	C₂×D₄×C₄₁	—	2	4	no	—
45	C₂×D₄×C₄₁	—	2	4	no	—
46	C₂×D₄×C₄₁	—	2	4	no	—
47	C₂×D₄×C₄₁	—	2	2	no	—
48	C₂×D₄×C₄₁	—	2	4	no	—
49	C₂×D₄×C₄₁	—	4	2	no	—
50	C₂×D₄×C₄₁	—	4	2	no	—
51	C₂×D₄×C₄₁	D₄	1	4	yes	complemented
52	C₂²×C₄×C₄₁	—	2	4	no	—
53	C₂²×C₄×C₄₁	—	2	2	no	—
54	C₂²×C₄×C₄₁	—	2	4	no	—
55	C₂²×C₄×C₄₁	—	4	2	no	—
56	C₂²×C₄×C₄₁	—	4	2	no	—
57	C₂²×C₄×C₄₁	—	4	2	no	—
58	C₂²×C₄×C₄₁	D₄	1	4	yes	complemented
59	C₂²×C₄×C₄₁	C₂³	1	2	yes	—
60	C₂²×C₄×C₄₁	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₄₁	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₄₁	C₂³	1	2	yes	—
63	C₂²×C₄×C₄₁	C₂³	1	1	yes	characteristic
64	C₂²×C₄×C₄₁	C₂³	1	1	yes	characteristic
65	C₂²×C₄×C₄₁	C₂³	1	1	yes	characteristic
66	C₂²×C₄×C₄₁	C₂³	1	2	yes	—
67	C₄⋊C₄×C₄₁	—	2	2	no	—
68	C₄⋊C₄×C₄₁	—	2	4	no	—
69	C₄⋊C₄×C₄₁	—	2	2	no	—
70	C₂²⋊C₄×C₄₁	—	2	2	no	—
71	C₂²⋊C₄×C₄₁	—	2	4	no	—
72	C₂²⋊C₄×C₄₁	—	2	2	no	—
73	C₂²⋊C₄×C₄₁	—	2	4	no	—
74	C₂²⋊C₄×C₄₁	—	2	4	no	—
75	C₂²⋊C₄×C₄₁	—	2	4	no	—
76	C₂²⋊C₄×C₄₁	—	4	2	no	—
77	C₄²×C₄₁	—	2	2	no	—
78	C₂³×C₄₁	—	2	2	no	—
79	C₂³×C₄₁	—	2	2	no	—
80	C₂³×C₄₁	—	2	1	no	—
81	C₂³×C₄₁	—	2	1	no	—
82	C₂³×C₄₁	—	2	2	no	—
83	C₂³×C₄₁	—	2	2	no	—
84	C₂³×C₄₁	—	2	2	no	—
85	C₂³×C₄₁	—	2	2	no	—
86	C₂³×C₄₁	—	2	2	no	—
87	C₂³×C₄₁	—	2	2	no	—
88	C₂³×C₄₁	—	2	2	no	—
89	C₂³×C₄₁	—	2	2	no	—
90	C₂³×C₄₁	—	2	2	no	—
91	C₂³×C₄₁	C₂×D₄	1	2	yes	—
92	C₂³×C₄₁	C₂×D₄	1	2	yes	—
93	C₂³×C₄₁	C₂×D₄	1	2	yes	—
94	C₂³×C₄₁	#41	1	2	yes	—
95	C₂³×C₄₁	C₂⁴	1	1	yes	characteristic
96	D₄×C₄₁	—	2	8	no	—
97	D₄×C₄₁	—	4	8	no	—
98	D₄×C₄₁	—	4	4	no	—
99	D₄×C₄₁	—	4	8	no	—
100	C₂×C₄×C₄₁	—	2	1	no	—
101	C₂×C₄×C₄₁	—	2	2	no	—
102	C₂×C₄×C₄₁	—	2	1	no	—
103	C₂×C₄×C₄₁	—	2	2	no	—
104	C₂×C₄×C₄₁	—	2	2	no	—
105	C₂×C₄×C₄₁	—	2	2	no	—
106	C₂×C₄×C₄₁	—	2	1	no	—
107	C₂×C₄×C₄₁	—	2	2	no	—
108	C₂×C₄×C₄₁	—	2	1	no	—
109	C₂×C₄×C₄₁	—	2	2	no	—
110	C₂×C₄×C₄₁	—	2	2	no	—
111	C₂×C₄×C₄₁	—	2	2	no	—
112	C₂×C₄×C₄₁	—	2	2	no	—
113	C₂×C₄×C₄₁	—	2	2	no	—
114	C₂×C₄×C₄₁	—	2	2	no	—
115	C₂×C₄×C₄₁	—	2	2	no	—
116	C₂×C₄×C₄₁	—	4	4	no	—
117	C₂×C₄×C₄₁	—	4	8	no	—
118	C₂×C₄×C₄₁	—	4	4	no	—
119	C₂×C₄×C₄₁	C₂×D₄	1	2	yes	—
120	C₂×C₄×C₄₁	C₂×D₄	1	2	yes	—
121	C₂×C₄×C₄₁	C₂×D₄	1	2	yes	—
122	C₂×C₄×C₄₁	#41	1	4	yes	—
123	C₂²×C₄₁	—	2	2	no	—
124	C₂²×C₄₁	—	2	2	no	—
125	C₂²×C₄₁	—	2	4	no	—
126	C₂²×C₄₁	—	2	4	no	—
127	C₂²×C₄₁	—	2	2	no	—
128	C₂²×C₄₁	—	2	4	no	—
129	C₂²×C₄₁	—	4	1	no	—
130	C₂²×C₄₁	—	4	1	no	—
131	C₂²×C₄₁	—	4	2	no	—
132	C₂²×C₄₁	—	4	2	no	—
133	C₂²×C₄₁	—	4	2	no	—
134	C₂²×C₄₁	—	4	2	no	—
135	C₂²×C₄₁	—	4	2	no	—
136	C₂²×C₄₁	—	4	2	no	—
137	C₂²×C₄₁	—	4	2	no	—
138	C₂²×C₄₁	—	4	2	no	—
139	C₂²×C₄₁	#121	1	4	yes	—
140	C₂²×C₄₁	C₂²×D₄	1	1	yes	characteristic
141	C₂²×C₄₁	C₂²×D₄	1	1	yes	characteristic
142	C₂²×C₄₁	C₂×#41	1	1	yes	characteristic
143	C₂²×C₄₁	C₂×#41	1	2	yes	—
144	C₂²×C₄₁	H(5,𝔽₂)	1	2	yes	—
145	C₄×C₄₁	—	2	2	no	—
146	C₄×C₄₁	—	2	2	no	—
147	C₄×C₄₁	—	2	2	no	—
148	C₄×C₄₁	—	2	4	no	—
149	C₄×C₄₁	—	4	2	no	—
150	C₄×C₄₁	—	4	1	no	—
151	C₄×C₄₁	—	4	1	no	—
152	C₄×C₄₁	—	4	2	no	—
153	#2543	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
154	C₂×C₄₁	—	2	4	no	—
155	C₂×C₄₁	—	4	2	no	—
156	C₂×C₄₁	—	8	2	no	—
157	C₂×C₄₁	C₂×#121	1	1	yes	characteristic
158	C₂×C₄₁	#532	1	1	yes	characteristic
159	C₂×C₄₁	#534	1	1	yes	characteristic
160	C₂×C₄₁	#546	1	2	yes	—
161	C₂×C₄₁	#553	1	2	yes	—
162	C₂×#121	C₂×C₄₁	1	1	yes	characteristic, complemented
163	C₂×#121	C₂×C₄₁	1	2	yes	complemented
164	C₂×C₄×D₄	C₂×C₄₁	1	1	yes	characteristic, complemented
165	#398	C₂×C₄₁	1	1	yes	characteristic, complemented
166	#397	C₂×C₄₁	1	1	yes	characteristic, complemented
167	#394	C₂×C₄₁	1	1	yes	characteristic, complemented
168	#392	C₂×C₄₁	1	1	yes	characteristic, complemented
169	#388	C₂×C₄₁	1	2	yes	complemented
170	#386	C₂×C₄₁	1	2	yes	complemented
171	#386	C₂×C₄₁	1	2	yes	complemented
172	C₂×#95	C₂×C₄₁	1	1	yes	characteristic, complemented
173	C₄₁	#2543	1	1	yes	characteristic, direct product factor, Sylow
174	C₂²×D₄	C₂²×C₄₁	1	2	yes	—
175	C₂²×D₄	C₂²×C₄₁	1	2	yes	—
176	C₂²×D₄	C₂²×C₄₁	1	1	yes	characteristic
177	C₂²×D₄	C₂²×C₄₁	1	1	yes	characteristic
178	C₂³×C₄	C₂²×C₄₁	1	2	yes	—
179	C₂³×C₄	C₂²×C₄₁	1	1	yes	characteristic
180	C₂³×C₄	C₂²×C₄₁	1	1	yes	characteristic
181	#121	—	2	8	no	—
182	#121	—	2	4	no	—
183	C₄×D₄	—	2	4	no	—
184	C₂×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
185	C₂×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
186	C₂×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
187	C₂×C₂²⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
188	C₂×C₂²⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
189	C₂×C₂²⋊C₄	C₂²×C₄₁	1	2	yes	—
190	C₂×C₂²⋊C₄	C₂²×C₄₁	1	2	yes	—
191	C₂×C₂²⋊C₄	C₂²×C₄₁	1	2	yes	—
192	C₂×C₂²⋊C₄	C₂²×C₄₁	1	2	yes	—
193	C₂×C₂²⋊C₄	C₂²×C₄₁	1	2	yes	—
194	C₂×C₄²	C₂²×C₄₁	1	1	yes	characteristic
195	#95	C₂²×C₄₁	1	2	yes	—
196	#95	C₂²×C₄₁	1	1	yes	characteristic
197	#95	C₂²×C₄₁	1	1	yes	characteristic
198	#95	C₂²×C₄₁	1	2	yes	—
199	#95	C₂²×C₄₁	1	2	yes	—
200	C₂⁴	C₂³×C₄₁	1	2	yes	—
201	C₂⁴	C₂³×C₄₁	1	1	yes	characteristic
202	C₂⁴	C₂³×C₄₁	1	1	yes	characteristic
203	C₂×D₄	—	2	2	no	—
204	C₂×D₄	—	2	2	no	—
205	C₂×D₄	—	2	4	no	—
206	C₂×D₄	—	2	4	no	—
207	C₂×D₄	—	2	4	no	—
208	C₂×D₄	—	2	4	no	—
209	C₂×D₄	—	2	2	no	—
210	C₂×D₄	—	2	4	no	—
211	C₂×D₄	—	4	2	no	—
212	C₂×D₄	—	4	2	no	—
213	C₂×D₄	D₄×C₄₁	1	4	yes	complemented
214	C₂²×C₄	—	2	4	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	2	4	no	—
217	C₂²×C₄	—	4	2	no	—
218	C₂²×C₄	—	4	2	no	—
219	C₂²×C₄	—	4	2	no	—
220	C₂²×C₄	D₄×C₄₁	1	4	yes	complemented
221	C₂²×C₄	C₂³×C₄₁	1	2	yes	—
222	C₂²×C₄	C₂³×C₄₁	1	1	yes	characteristic
223	C₂²×C₄	C₂³×C₄₁	1	1	yes	characteristic
224	C₂²×C₄	C₂³×C₄₁	1	2	yes	—
225	C₂²×C₄	C₂³×C₄₁	1	1	yes	characteristic
226	C₂²×C₄	C₂³×C₄₁	1	1	yes	characteristic
227	C₂²×C₄	C₂³×C₄₁	1	1	yes	characteristic
228	C₂²×C₄	C₂³×C₄₁	1	2	yes	—
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	—	2	4	no	—
231	C₄⋊C₄	—	2	2	no	—
232	C₂²⋊C₄	—	2	2	no	—
233	C₂²⋊C₄	—	2	4	no	—
234	C₂²⋊C₄	—	2	2	no	—
235	C₂²⋊C₄	—	2	4	no	—
236	C₂²⋊C₄	—	2	4	no	—
237	C₂²⋊C₄	—	2	4	no	—
238	C₂²⋊C₄	—	4	2	no	—
239	C₄²	—	2	2	no	—
240	C₂³	—	2	2	no	—
241	C₂³	—	2	2	no	—
242	C₂³	—	2	1	no	—
243	C₂³	—	2	1	no	—
244	C₂³	—	2	2	no	—
245	C₂³	—	2	2	no	—
246	C₂³	—	2	2	no	—
247	C₂³	—	2	2	no	—
248	C₂³	—	2	2	no	—
249	C₂³	—	2	2	no	—
250	C₂³	—	2	2	no	—
251	C₂³	—	2	2	no	—
252	C₂³	—	2	2	no	—
253	C₂³	C₂×D₄×C₄₁	1	2	yes	—
254	C₂³	C₂×D₄×C₄₁	1	2	yes	—
255	C₂³	C₂×D₄×C₄₁	1	2	yes	—
256	C₂³	#41×C₄₁	1	2	yes	—
257	C₂³	C₂⁴×C₄₁	1	1	yes	characteristic
258	D₄	—	2	8	no	—
259	D₄	—	4	8	no	—
260	D₄	—	4	4	no	—
261	D₄	—	4	8	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	4	8	no	—
280	C₂×C₄	—	4	4	no	—
281	C₂×C₄	C₂×D₄×C₄₁	1	2	yes	—
282	C₂×C₄	C₂×D₄×C₄₁	1	2	yes	—
283	C₂×C₄	C₂×D₄×C₄₁	1	2	yes	—
284	C₂×C₄	#41×C₄₁	1	4	yes	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	2	4	no	—
288	C₂²	—	2	4	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	4	no	—
291	C₂²	—	4	1	no	—
292	C₂²	—	4	1	no	—
293	C₂²	—	4	2	no	—
294	C₂²	—	4	2	no	—
295	C₂²	—	4	2	no	—
296	C₂²	—	4	2	no	—
297	C₂²	—	4	2	no	—
298	C₂²	—	4	2	no	—
299	C₂²	—	4	2	no	—
300	C₂²	—	4	2	no	—
301	C₂²	#121×C₄₁	1	4	yes	—
302	C₂²	C₂²×D₄×C₄₁	1	1	yes	characteristic
303	C₂²	C₂²×D₄×C₄₁	1	1	yes	characteristic
304	C₂²	C₂×#41×C₄₁	1	1	yes	characteristic
305	C₂²	C₂×#41×C₄₁	1	2	yes	—
306	C₂²	H(5,𝔽₂)×C₄₁	1	2	yes	—
307	C₄	—	2	2	no	—
308	C₄	—	2	2	no	—
309	C₄	—	2	2	no	—
310	C₄	—	2	4	no	—
311	C₄	—	4	2	no	—
312	C₄	—	4	1	no	—
313	C₄	—	4	1	no	—
314	C₄	—	4	2	no	—
315	C₂	—	2	4	no	—
316	C₂	—	4	2	no	—
317	C₂	—	8	2	no	—
318	C₂	C₂×#121×C₄₁	1	1	yes	characteristic
319	C₂	C₄₁×#532	1	1	yes	characteristic
320	C₂	C₄₁×#534	1	1	yes	characteristic
321	C₂	C₄₁×#546	1	2	yes	—
322	C₂	C₄₁×#553	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₄₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₄₁
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×#121×C₄₁	C₂	semidirect
2. C₂×#121×C₄₁	C₂	semidirect
3. C₂×C₄×D₄×C₄₁	C₂	semidirect
4. C₄₁×#398	C₂	semidirect
5. C₄₁×#397	C₂	semidirect
6. C₄₁×#394	C₂	semidirect
7. C₄₁×#392	C₂	semidirect
8. C₄₁×#388	C₂	semidirect
9. C₄₁×#386	C₂	semidirect
10. C₄₁×#386	C₂	semidirect
11. C₂×#95×C₄₁	C₂	semidirect
51. C₂×D₄×C₄₁	D₄	semidirect
58. C₂²×C₄×C₄₁	D₄	semidirect
153. #2543	C₄₁	direct (N × H)
162. C₂×#121	C₂×C₄₁	semidirect
163. C₂×#121	C₂×C₄₁	semidirect
164. C₂×C₄×D₄	C₂×C₄₁	semidirect
165. #398	C₂×C₄₁	semidirect
166. #397	C₂×C₄₁	semidirect
167. #394	C₂×C₄₁	semidirect
168. #392	C₂×C₄₁	semidirect
169. #388	C₂×C₄₁	semidirect
170. #386	C₂×C₄₁	semidirect
171. #386	C₂×C₄₁	semidirect
172. C₂×#95	C₂×C₄₁	semidirect
173. C₄₁	#2543	direct (N × H)
213. C₂×D₄	D₄×C₄₁	semidirect
220. C₂²×C₄	D₄×C₄₁	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#121×C₄₁	2
C₂×C₄×D₄×C₄₁	2
C₄₁×#398	2
C₄₁×#397	2
C₄₁×#394	2
C₄₁×#392	2
C₄₁×#388	2
C₄₁×#386	2
C₂×#95×C₄₁	2
#2543	41

Subgroups — C41×#2543 #19106323

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#2543 #19106323